- Bracket (mathematics)
In

mathematics , various typographical forms ofbracket s are frequently used inmathematical notation such as parentheses ( ), square brackets [ ] , curly brackets { }, and angle brackets < >. In the typical use, a mathematical expression is enclosed between an "opening bracket" and a matching "closing bracket". Generally such bracketing denotes some form of grouping: in evaluating an expression containing a bracketed sub-expression, the operators in the sub-expression take precedence over those surrounding it. Additionally, there are several specific uses and meanings for the various brackets.Historically, other notations, such as the vinculum, were similarly used for grouping; in present-day use, these notations have all specific meanings.

In the Z

formal specification language, curly braces denote a set and angle brackets denote a sequence.**ymbols for representing angle brackets**A variety of different symbols are used to represent angle brackets. In e-mail and other

ASCII text it is common to use the less-than (<) and greater-than (>) signs to represent angle brackets.Unicode has three pairs of dedicated characters: U+2329 (〈) and U+232A (〉) ("left/right-pointing angle bracket"), U+27E8 (⟨) and U+27E9 (⟩) ("mathematical left/right angle bracket"), and U+3008 (〈) and U+3009 (〉) ("left/right angle bracket"). InLaTeX the markup is langle and angle: $langle\; ,,\; angle,$.**Algebra**In

elementary algebra parentheses, ( ), are used to specify theorder of operations , terms inside the bracket are evaluated first, hence 2×(3 + 4) is 14 and (2×3) + 4 is 10. This notation is extended cover more generalalgebra involving variables: for example $(x+y)\; imes(x-y)$.Also in

mathematical expression s in general, parentheses are used to indicate grouping (that is, which parts belong together) when necessary to avoid ambiguities, or for the sake of clarity. For example, in the formula (εη)_{"X"}= ε_{"X"}η_{"X"}, used in the definition of composition of twonatural transformation s, the parentheses around εη serve to indicated that the indexing by "X" is applied to the composition εη, and not just its last component η.**Functions**The arguments to a function are frequently surrounded by brackets: $f(x)$. It is common to omit the parentheses around the argument when there is little chance of ambiguity, thus: $sin\; x$.

**Coordinates and vectors**In the

cartesian coordinate system brackets are used to specify the coordinates of a point: (2,3) denotes the point with "x"-coordinate 2 and "y"-coordinate 3.The

inner product of two vectors is commonly written as $langle\; a,\; b\; angle$, but the notation ("a", "b") is also used.**Intervals**Both parentheses, ( ), and square brackets, [ ] , can also be used to denote an interval. The notation

[a, c) is used to indicate an interval from a to c that is inclusive of a but exclusive of c. That is,[5, 12) would be the set of all real numbers between 5 and 12, including 5 but not 12. The numbers may come as close as they like to 12, including 11.999 and so forth (with any finite number of 9s), but 12.0 is not included. In Europe, the notation[5,12 [ is also used for this.The endpoint adjoining the square bracket is known as "closed", while the endpoint adjoining the parenthesis is known as "open". If both types of brackets are the same, the entire interval may be referred to as "closed" or "open" as appropriate. Whenever

infinity or negative infinity is used as an endpoint in the case of intervals on the real number line, it is always considered "open" and adjoined to a parenthesis. The endpoint can be closed when considering intervals on theextended real number line .**ets and groups**Curly brackets { } are used to identify the elements of a set: {"a","b","c"} denotes a set of three elements.

Angle brackets are used in

group theory to writegroup presentation s, and to denote the subgroup generated by a collection of elements.**Matrices**An explicitly given matrix is commonly written between large round or square brackets:

:$egin\{pmatrix\}a\; b\; \backslash c\; d\; end\{pmatrix\}quadquadegin\{bmatrix\}a\; b\; \backslash c\; d\; end\{bmatrix\}$

**Derivatives**The notation:$f^\{(n)\}(x),$stands for the "n"-th derivative of function "f", applied to argument "x". So, for example, if "f"("x") = exp(λ"x"), "f"

^{("n")}("x") = λ^{"n"}exp(λ"x"). This is to be contrasted with "f"^{"n"}("x") = "f"("f"(...("f"(x))...)), the "n"-fold application of "f" to argument "x".**Falling and rising factorial**The notation ("x")

_{"n"}is used to denote the "falling factorial ", an "n"-th degreepolynomial defined by:$(x)\_n=x(x-1)(x-2)cdots(x-n+1)=frac\{x!\}\{(x-n)!\}.$Confusingly, the same notation may be encountered as representing the "rising factorial", also called "

Pochhammer symbol ". Another notation for the same is "x"^{("n")}. It can be defined by:$x^\{(n)\}=x(x+1)(x+2)cdots(x+n-1)=frac\{(x+n-1)!\}\{(x-1)!\}.$**Quantum mechanics**In

quantum mechanics , angle brackets are also used as part of Dirac's formalism,bra-ket notation , to note vectors from thedual space s of the Bra .In statistical mechanics, angle brackets denote ensemble or time average.

**Lie bracket and commutator**In

group theory andring theory , square brackets are used to denote thecommutator . In group theory, the commutator[ "g","h"] is commonly defined as "g"^{−1}"h"^{−1}"gh". In ring theory, the commutator[ "a","b"] is defined as "ab" − "ba". Furthermore, in ring theory, braces are used to denote the anticommutator where {"a","b"} is defined as "ab" + "ba".The

**Lie bracket**of aLie algebra is abinary operation denoted by [·, ·] : $mathfrak\{g\}$ × $mathfrak\{g\}$ → $mathfrak\{g\}$. By using the commutator as a Lie bracket, every associative algebra can be turned into a Lie algebra. There are many different forms of**Lie bracket**, in particular theLie derivative and theJacobi-Lie bracket .**Floor / Ceiling functions and fractional part**Square brackets, as in nowrap|1= [π] = 3, are sometimes used to denote the

floor function , which rounds a real number down to the next integer.However the floor and ceiling functions are usually typeset with left and right square brackets where the upper (for floor function) or lower (for ceiling function) horizontal bars are missing, as in nowrap|1= ⌊π⌋ = 3 or nowrap|1= ⌈π⌉ = 4.Curly brackets, as in nowrap|1= {π} <

^{1}/_{7}, may denote thefractional part of a real number.**ee also***

Iverson bracket

*Algebraic bracket

*Binomial coefficient

*Poisson bracket

*Bracket polynomial

*Pochhammer symbol

*Frölicher-Nijenhuis bracket

*Nijenhuis-Richardson bracket

*Schouten-Nijenhuis bracket

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